2 definitions for argument, why?

In summary, there are two definitions for the argument in the wiki. The first one is given by the inverse of \mathrm{cis}\,\theta = \cos \theta + i \sin \theta = e^{i\theta}, which only applies when |z| = 1. The second one is given by \arg(z) = \ln(\sqrt[2 i]{z \div \bar{z} }) = \frac{ln(z) - ln(\bar{z})}{2 i}, which applies for any z \neq 0. These two definitions are related and can be used interchangeably, depending on the situation and the branch chosen for z^{1/(2i)}.
  • #1
Jhenrique
685
4
In the wiki, I found this definition for the argument:

c3b660fcef985a8d0781f6bfb6659a76.png


http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Exponential_definitions

However, in other page of the wiki (http://en.wikipedia.org/wiki/Complex_conjugate#Use_as_a_variable), I found this definition for argument:[tex]\arg(z) = \ln(\sqrt[2 i]{z \div \bar{z} }) = \frac{ln(z) - ln(\bar{z})}{2 i}[/tex]I don't understand why exist 2 defitions for the argument and how those 2 defitions are related.
 
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  • #2
Jhenrique said:
In the wiki, I found this definition for the argument:

c3b660fcef985a8d0781f6bfb6659a76.png
This gives the inverse of [itex]\mathrm{cis}\,\theta = \cos \theta + i \sin \theta = e^{i\theta}[/itex]. It is not a definition of the argument, but reflects the fact that if [itex]z = e^{i\theta}[/itex] then
[tex]
-i \log e^{i\theta} = -i(i \theta) = \theta = \arg z.
[/tex]
It doesn't give [itex]\arg z[/itex] if [itex]|z| = R \neq 1[/itex]:
[tex]
-i \log (Re^{i\theta}) = -i \log R + \theta \neq \arg z
[/tex]

However, in other page of the wiki (http://en.wikipedia.org/wiki/Complex_conjugate#Use_as_a_variable), I found this definition for argument:[tex]\arg(z) = \ln(\sqrt[2 i]{z \div \bar{z} }) = \frac{ln(z) - ln(\bar{z})}{2 i}[/tex]I don't understand why exist 2 defitions for the argument and how those 2 defitions are related.

This gives [itex]\arg z[/itex] for any [itex]z \neq 0[/itex] (if you choose the correct branch of [itex]z^{1/(2i)}[/itex]).
 
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  • #3
I liked your answer!
 
  • #4
There is almost always an alternative way of expressing the same mathematical argument, with a little imagination. It's not always obvious.
 
  • #5


I can explain that the two definitions for argument are related to different mathematical concepts and are used for different purposes. The first definition, found in the list of trigonometric identities, is related to the concept of complex numbers and their representation in polar form. In this context, the argument represents the angle between the complex number and the positive real axis.

The second definition, found in the page on complex conjugates, is related to the concept of logarithms and their use in solving equations involving complex numbers. In this context, the argument represents the phase angle of the complex number and is used to find the complex conjugate of a number.

It is important to note that both definitions are valid and useful in their respective contexts. The first definition is commonly used in trigonometry and calculus, while the second definition is commonly used in complex analysis and engineering.

Overall, the existence of two definitions for the argument highlights the versatility and applicability of complex numbers in various fields of mathematics and science. It is not uncommon for different definitions or representations of a concept to coexist and be used in different contexts.
 

Related to 2 definitions for argument, why?

What is an argument in science?

An argument in science refers to a logical and evidence-based explanation or hypothesis that is supported by data and research. It is used to explain, predict, or understand a phenomenon or event.

Why is it important to have multiple definitions for an argument in science?

Having multiple definitions for an argument in science allows for a more comprehensive understanding of a topic or concept. It also encourages critical thinking and analysis by considering different perspectives and evidence.

How do scientists use arguments in their research?

Scientists use arguments to support their theories and hypotheses. They gather and analyze data, conduct experiments, and make observations to build a logical and evidence-based argument in support of their findings.

Can arguments in science change over time?

Yes, arguments in science can change over time as new evidence and research becomes available. As scientists continue to gather data and conduct experiments, their understanding and interpretation of a topic or concept may evolve, leading to changes in their arguments.

Are there limitations to using arguments in science?

While arguments in science provide a valuable tool for understanding and explaining phenomena, they are not infallible. They may be limited by the quality of the evidence, biases of the researcher, or incomplete understanding of a topic. It is important for scientists to continually evaluate and reassess their arguments in light of new information.

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